Let $g$ be a function defined for all real numbers except for $x=2$.
Also let $g^{\prime}$, the derivative of $g$, be defined as $g^{\prime}(x)=\frac{x^{2}}{(x-2)^{3}}$.
On which intervals is $g$ increasing?
Choose 1 answer:
(A) $(-\infty, 0)$ and $(0,2)$
(B) $(-\infty, 0)$ and $(2, \infty)$
(C) $(2, \infty)$ only
(D) $(0,2)$ only
(E) The entire domain of $g$

Question

Let $g$ be a function defined for all real numbers except for $x=2$.
Also let $g^{\prime}$, the derivative of $g$, be defined as $g^{\prime}(x)=\frac{x^{2}}{(x-2)^{3}}$.
On which intervals is $g$ increasing?
Choose 1 answer:
(A) $(-\infty, 0)$ and $(0,2)$
(B) $(-\infty, 0)$ and $(2, \infty)$
(C) $(2, \infty)$ only
(D) $(0,2)$ only
(E) The entire domain of $g$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Determine the intervals where $g'(x) = \frac{x^2}{(x - 2)^3}$ is greater than zero. ‖ ‖ step 2 ⋮ From the asksia-ll calculation list, the solution to $g'(x) > 0$ is $x > 2$. ‖ ‖ step 3 ⋮ Since $g'(x) > 0$ for $x > 2$, the function $g$ is increasing on the interval $(2, \infty)$. ‖ ∻Answer∻ ⚹ (C) $(2, \infty)$ only ⚹ ∻Key Concept∻ ⚹Intervals of Increase⚹ ∻Explanation∻ ⚹A function is increasing on intervals where its derivative is positive. The asksia-ll calculator determined that $g'(x) > 0$ for $x > 2$, so $g$ is increasing on the interval $(2, \infty)$.⚹◊What is the relationship between the sign of $g'(x)$ and the increasing/decreasing behavior of the function $g$?⍭ Generate me a similar question◊

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